And what the relationship between this red circle and the blue circle is. A parametric representation of critical curves and. Nonparametric equations can be explicit or implicit. As t varies, the point x, y ft, gt varies and traces out a curve c, which we call a parametric curve. Parametric representation of a surface pencil with a common. Implicit and explicit forms are often referred to as nonparametric forms. Parametric curves brian curless cse 457 spring 2015 2 reading required. Curves having parametric form are called parametric curves. Parametric curves in this section we explore the use of a parameter t in the description of curves. In this section we will introduce parametric equations and parametric curves i. Pdf implicit representation of parametric curves and surfaces.
For instance, in tracking the movement of a satellite, we would naturally want to give its location in terms of time. Parametric curves in the past, we mostly worked with curves in the form y fx. In parametric representation of curves and surfaces, it is possible to. We shall see that it has some advantages over the more usual cartesian description.
Many products need freeform, or synthetic curved surfaces. In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Download fulltext pdf implicit representation of parametric curves and surfaces article pdf available in computer vision graphics and image processing 281. Thus, compact representations of implicit curves allow the gpu to process more image data at runtime to render a more rich set of images on a computer display. Parametric representation of a surface pencil with a. Parametric vs nonparametric models parametric models assume some. These functions yield a parametric representation of the functional relationship between x and y, and equations are said to be parametric equations of the corresponding curve. Parametric equations of lines general parametric equations in this part of the unit we are going to look at parametric curves. Such a function is the explicit representation of the curve.
In the twodimensional coordinate system, parametric equations are useful for describing curves that are not necessarily functions. Then, are parametric equations for a curve in the plane. On the basis of this, we present a parametric representation of such curves. Piecing together hermite curves its easy to make a multisegment hermite spline each piece is specified by a cubic hermite curve just specify the position and tangent at each joint the pieces fit together with matched positions and first derivatives gives c1 continuity. Like curves, surfaces can be represented in implicit, explicit, or parametric form. Parametric curve with an implicit domain springerlink.
Reparameterization parameterizations are in general not unique. Using the frenet frame of the given asymptotic curve, we present the surface as a linear combination of this frame and analyse the necessary and. A parametric representation of critical curves and caustics. Parametric curves curves and surfaces can have explicit, implicit, and parametric representations. Representation of curves and surfaces we first introduce three forms to represent geometric objects mathematically.
Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization alternatively. In the case of two variables x and y, the expression fx, y 0 can be geometrically interpreted as the equation of a plane curve. A particular bezier curve is set up by its parametric representation, let b. The parametric representation of sm by means of the lowner differential equation is almost the same as of s. Us20060095489a1 representation of implicit curves for. In nurbs curves the knot values do not have to be uniformly spaced. The vertex shader supports local frenetserret frames approximations and more robust parallel transport frames at expense of performance. Parametric equations introduction, eliminating the paremeter t, graphing plane curves, precalculus duration. Nurbs have a weighting factor hi associated with each control point. Length of a curve calculus with parametric equations let cbe a parametric curve described by the parametric equations x ft. Other articles where parametric curve is discussed. In three dimensions you will also have z z 0 st in two dimensions a line can be described as the solution to one linear equation, and in three dimensions as the solution to two such equations. The explicit representation is not general, since it cannot represent vertical lines and is also singlevalued. Compact and accurate piecewise parametric representations of implicit functions may be achieved by iteratively selecting ranges of parameterizing regions and testing each for satisfying an intervalized super convergence test.
Parametric curves cse 457 winter 2014 2 reading required. Parametric representation of synthetic curves free pdf note free download, computer aided design pdf notes. So the complexity of the model is bounded even if the amount of data is unbounded. Parametric representation a parametric representation of a function expresses the functional relationship between several variables by means of auxiliary variable parameters. Parametric representations 3 basic representation strategies. Now we will look at parametric equations of more general trajectories.
Projectile motion sketch and axes, cannon at origin, trajectory mechanics gives and. Parametric representation of a surface pencil with an asymptotic curve suppose we are given a 3dimensional parametric curve r s, l 1. They are the parametric, implicit and explicit forms. Parametric representation of curves the position of a particle moving in threedimensional space is often speci. Fast animated curve rendering on the gpu with parametric equations. Curves in modeling curves produced in modeling systems. Parametric representation for 2d curves curves can be represented parametrically as. A mathematical function describes the path of a curve parametric techniques. We shall see that this is, in some ways, far more useful than either the cartesian description or the polar form. To this point in both calculus i and calculus ii weve looked almost exclusively at functions in the form \y f\left x \right\ or \x h\left y \right\ and almost all of the formulas that weve developed require that functions be in one of these two forms. Write the equation for a parametric cubic polynomial curve in matrix. Curves defined by parametric equations mathematics.
Calculus iii parametric surfaces pauls online math notes. Parametric representation an overview sciencedirect topics. Curvature extrema of planar parametric polynomial cubic curves. The present result is helpful for efficiently constructing many templates of light curves due to binary systems, particularly extrasolar planets, which cause spikes in the light curves when a source. However, this format does not encompass all the curves one encounters in applications. This is an example of an explicit non parametric curve form.
And i should maybe say oneparameter parametric function. An introduction to splines for use in computer graphics and geometric modeling, 1987. Parametric representation article about parametric. The parameter is an independent variable that both \x\ and \y\ depend on, and as the parameter increases, the values of \x\ and \y\ trace out a path along a plane curve. In this paper we present a new representation of curve, named parametric curve with an implicit domain pcid, which is a curve that exists in. A surface model represents the skin of an object, these skins have no thickness or material type. Ordinarily, the curves or surfaces are restricted in the literature to a domain. Parametric representations are the most common in computer graphics. In one aspect, the implicit function is represented as a compact form of one or more representations of such convergence regions. Nurbs curves are useful because they allow exact representation of conic curves. Although we shall only study planar curves curves lying in a.
Curves defined by parametric equations each value of t determines a point x, y, which we can plot in a coordinate plane. Picture a function in 2d space, it is a curve instead of a plane. Here, we do not so restrict parametric curves and surfaces. Surface modeling surface modeling surface models define the surface features, as well as the edges, of objects. The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the. The approach to sketching the curve is straightforward. Straight lines conic sections freeform parametric curves bsplines, nurbs curves of intersection between surfaces defined during model construction curve use in design. Computer graphics parametric curves 2005 fabio pellacini 5 parametric representation of splines general parametric curve can be written as. The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter. Giventheparameters, future predictions, x, are independent of the observed data, d. Consider the following parameterizations for a line. A straight line with slope r through the point x 0, y 0 can be represented parametrically as y y 0 rt, x x 0 t. We shall apply the methods for cartesian coordinates to. In this section we will take a look at the basics of representing a surface with parametric equations.
Implicit representation of parametric curves and surfaces. Curve representation all forms of geometric modeling require the ability to define curves. Parametric curves cs527 computer graphics 1 note 9. Parametric curves general parametric equations we have seen parametric equations for lines. Mathematica demonstrations, parametric curve and surface, architectural form. The simple idea of caratheodory was realized in the parametric method owing to lowners skill in describing a piecewise smooth deformation wz, t, 0. Calculus with parametric equationsexample 2area under a curvearc length. And parameter is just kind of a fancy word for input. For each value of x, only a single value of y is normally computed by the function. We will also see how the parameterization of a surface can be used to find a normal vector for the surface which will be very useful in a couple of sections and how the parameterization can be used to find the surface area of a surface. This is simply the idea that a point moving in space traces out a path over time. Repeating what was said earlier, a parametric curve is simply the idea that a point moving in the space traces out a path.
The caustics can also be expressed with the same parameterization. We can represent any functions of curve curved surface using parametric equation. Pdf implicit representation of parametric curves and. Parametric equations can be used to describe all types of curves that can be represented on a plane but are most often used in situations where curves on. Parametric curves curve representation curves can be described mathematically by nonparametric or parametric equations. Thus there are four variables to consider, the position of the point x,y,z and an independent variable t, which we can think of as time. Pdf scalar and parametric splines curves and surfaces. Sometimes and are given as functions of a parameter. To this point in both calculus i and calculus ii weve looked almost exclusively at functions in the form \y f\left x \right\ or \x h\left y \right\ and almost all of the formulas that weve developed require that functions be.
Nurbs curves nurbs means nonuniform rational bspline. But here i just kind of want to give an intuition for what parametric surfaces are all about, how its a way of visualizing something that has a twodimensional input and a threedimensional output. Parametric representation of synthetic curves analytic curves are usually not sufficient to meet geometric design requirements of mechanical parts. Among other things, a compact representation of implicit curves reduces the computational costs of processing needed by the gpu to render images comprising implicit curves. Curves can be described mathematically by nonparametric or parametric equations.
This representation can be extended to get a parametric representation for complex surfaces. Aug 18, 2003 on the basis of this, we present a parametric representation of such curves. We obtain the parametric representation for a surface pencil whose members have the same curve as a given asymptotic curve. Geometric modeling, triangular mesh, parametric curve, quaternion curve, spherical parameterization. Px,dpx therefore capture everything there is to know about the data. Now make it a function of 2 variables and you can create a solid 2d object. For a nonparametric curve, the coordinates y and z of a point on the curve are expressed as two separate functions of the third coordinate x as the independent variable. Parametric representation of synthetic curves free pdf note. And what makes it a parametric function is that we think about it as drawing a curve and its output is multidimensional.
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